We prove that every reversible Markov semigroup which satisfies a Poincare inequality satisfies a matrix-valued Poincare inequality for Hermitian $dtimes d$ matrix valued functions, with the same Poincare constant. This generalizes recent results [Aoun et al. 2019, Kathuria 2019] establishing such inequalities for specific semigroups and consequently yields new matrix concentration inequalities. The short proof follows from the spectral theory of Markov semigroup generators.
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